Method and system for using fitted relaxation data to improve a product

Abstract

A system to improve a product based on a relaxation response includes a memory configured to store relaxation response data of a sample. The relaxation response data includes time data and amplitude data. A processor is operatively coupled to the memory and configured to convert the relaxation response data to linear-amplitude versus log-time data. The processor also performs a least-squares fit of the converted relaxation response data to a heavy-tail function to determine one or more fit parameter values. The processor also updates a design for the sample based at least in part on the one or more fit parameter values.

What is claimed is: 
     
         1 . A system to improve a product based on a relaxation response, the system comprising:
 a memory configured to store relaxation response data of a sample, wherein the relaxation response data includes time data and amplitude data; and   a processor operatively coupled to the memory and configured to:
 convert the relaxation response data to linear-amplitude versus log-time data; 
 perform a least-squares fit of the converted relaxation response data to a heavy-tail function to determine one or more fit parameter values; and 
 update a design for the sample based at least in part on the one or more fit parameter values. 
   
     
     
         2 . The system of  claim 1 , wherein the processor is further configured to determine an offset parameter for the relaxation response data. 
     
     
         3 . The system of  claim 2 , wherein the processor is configured to:
 determine a mean value of the relaxation response data; and   shift the relaxation response data relative to the mean value to create a dataset that has a zero mean value, wherein the offset parameter is based on the created dataset.   
     
     
         4 . The system of  claim 1 , wherein the processor is configured to generate an error estimate for each of the one or more fit parameter values. 
     
     
         5 . The system of  claim 4 , wherein to generate the error estimate of a fit parameter, the processor transforms the fit parameter to a space in which variance of a fit of the fit parameter is quadratic. 
     
     
         6 . The system of  claim 4 , wherein the processor is configured to generate a confidence interval for each of the one or more fit parameter values based at least in part on the error estimate. 
     
     
         7 . The system of  claim 6 , wherein the processor applies a Hessian analysis function to the error estimate to generate the confidence interval. 
     
     
         8 . The system of  claim 1 , wherein the processor is configured to generate a report that includes the one or more fit parameter values and one or more confidence intervals associated with the one or more fit parameter values. 
     
     
         9 . The system of  claim 1 , wherein the one or more fit parameter values includes a value of a time scale of relaxation of the sample. 
     
     
         10 . The system of  claim 1 , wherein the one or more fit parameter values includes a value of an amplitude of relaxation of the sample. 
     
     
         11 . The system of  claim 1 , wherein the one or more fit parameter values includes a value of a molecularity ratio of an initial minority concentration of the sample to a majority concentration of the sample. 
     
     
         12 . The system of  claim 1 , wherein the one or more fit parameter values includes a value of an anomalous diffusion exponent for the sample. 
     
     
         13 . The system of  claim 1 , further comprising an excitation device that excites the sample such that the sample exhibits the relaxation response that is a source of the relaxation response data. 
     
     
         14 . A method comprising:
 storing, in a memory of a computing system, relaxation response data of a sample, wherein the relaxation response data includes time data and amplitude data;   converting, by a processor of the computing system, the relaxation response data to linear-amplitude versus log-time data;   performing, by the processor, a least-squares fit of the converted relaxation response data to a heavy-tail function to determine one or more fit parameter values; and   updating a design for the sample based at least in part on the one or more fit parameter values.   
     
     
         15 . The method of  claim 14 , further comprising:
 determining, by the processor, a mean value of the relaxation response data;   shifting the relaxation response data relative to the mean value to create a dataset that has a zero mean value; and   determining an offset parameter for the relaxation response data based on the created dataset.   
     
     
         16 . The method of  claim 14 , further comprising generating, by the processor, an error estimate for each of the one or more fit parameter values. 
     
     
         17 . The method of  claim 16 , further comprising generating, by the processor, a confidence interval for each of the one or more fit parameter values based at least in part on the error estimate. 
     
     
         18 . The method of  claim 17 , further comprising applying, by the processor, a Hessian analysis function to the error estimate to generate the confidence interval. 
     
     
         19 . The method of  claim 14 , further comprising generating, by the processor, a report that includes the one or more fit parameter values and one or more confidence intervals associated with the one or more fit parameter values. 
     
     
         20 . The method of  claim 14 , further comprising exciting, by an excitation device in communication with the computing system, the sample such that the sample exhibits a relaxation response that is a source of the relaxation response data.